Abstract
This paper investigates the existence and uniqueness of solutions to neutral stochastic functional differential equations with pure jumps (NSFDEwPJs). The boundedness and almost sure exponential stability are also considered. In general, the classical existence and uniqueness theorem of solutions can be obtained under a local Lipschitz condition and linear growth condition. However, there are many equations that do not obey the linear growth condition. Therefore, our first aim is to establish new theorems where the linear growth condition is no longer required whereas the upper bound for the diffusion operator will play a leading role. Moreover, the pth moment boundedness and almost sure exponential stability are also obtained under some loose conditions. Finally, we present two examples to illustrate the effectiveness of our results.
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References
Wu L, Zheng W X, Gao H. Dissipativity-based sliding mode control of switched stochastic systems. IEEE Trans Automat Contr, 2013, 58: 785–791
Zhang B, Deng F Q, Zhao X Y, et al. Hybrid control of stochastic chaotic system based on memristive Lorenz system with discrete and distributed time-varying delays. IET Contr Theor Appl, 2016, 10: 1513–1523
Wu D, Luo X, Zhu S. Stochastic system with coupling between non-Gaussian and Gaussian noise terms. Phys A-Stat Mech Appl, 2007, 373: 203–214
Shao J, Yuan C. Transportation-cost inequalities for diffusions with jumps and its application to regime-switching processes. J Math Anal Appl, 2015, 425: 632–654
Elliott R J, Osakwe C J U. Option pricing for pure jump processes with markov switching compensators. Finance Stochast, 2006, 10: 250–275
Lee S S, Mykland P A. Jumps in financial markets: a new nonparametric test and jump dynamics. Rev Financ Stud, 2008, 21: 2535–2563
Mao W, Zhu Q, Mao X. Existence, uniqueness and almost surely asymptotic estimations of the solutions to neutral stochastic functional differential equations driven by pure jumps. Appl Math Comput, 2015, 254: 252–265
Agarwal R P. Editorial announcement. J Inequal Appl, 2011, 2011: 1
Song M H, Hu L J, Mao X R, et al. Khasminskii-type theorems for stochastic functional differential equations. Discrete Cont Dyn Syst - Ser B, 2013, 18: 1697–1714
Luo Q, Mao X, Shen Y. Generalised theory on asymptotic stability and boundedness of stochastic functional differential equations. Automatica, 2011, 47: 2075–2081
Wu F, Hu S. Khasminskii-type theorems for stochastic functional differential equations with infinite delay. Stat Probab Lett, 2011, 81: 1690–1694
Mao X, Rassias M J. Khasminskii-type theorems for stochastic differential delay equations. Stochastic Anal Appl, 2005, 23: 1045–1069
Applebaum D. Lévy Processes and Stochastic Calculus. Cambridge: Cambridge University Press, 2009
Mao X R. Stochastic Differential Equations and Applications. 2nd ed. Cambridge: Woodhead Publishing, 2008
Meyer P A. Probability and Potentials. Waltham: Blaisdell, 1966
Paw lucki W, Plésniak W. Markov’s inequality and C∞ functions on sets with polynomial cusps. Math Ann, 1986, 275: 467–480
Beckner W. Inequalities in fourier analysis. Ann Math, 1975, 102: 159–182
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant Nos. 61573156, 61273126, 61503142), the Ph.D. Start-up Fund of Natural Science Foundation of Guangdong Province (Grant No. 2014A030310388), and Fundamental Research Funds for the Central Universities (Grant No. x2zdD2153620).
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Li, M., Deng, F. & Mao, X. Basic theory and stability analysis for neutral stochastic functional differential equations with pure jumps. Sci. China Inf. Sci. 62, 12204 (2019). https://doi.org/10.1007/s11432-017-9302-9
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DOI: https://doi.org/10.1007/s11432-017-9302-9